Ethylene-vinyl alcohol copolymers (EVOH) have been known to have excellent gas barrier properties, but have drawbacks in that they have poor flexibility, and particularly they are brittle at low temperature, whereas polyamide resins have been known to have excellent fatigue durability as well as excellent gas barrier properties. Although it is expected that a thermoplastic elastomer comprising as a continuous phase a thermoplastic resin composition comprising an ethylene-vinyl alcohol copolymer and a polyamide resin as well as a modified elastomer as a dispersed phase has excellent gas barrier properties and fatigue durability as described above, there is a problem that, when an ethylene-vinyl alcohol copolymer, a polyamide resin and a modified elastomer are kneaded to produce such a thermoplastic elastomer composition, the reaction of the ethylene-vinyl alcohol copolymer with the polyamide resin results in gelation. Although a method of capping the end amino groups of a polyamide resin capable of reacting with an ethylene-vinyl alcohol copolymer with an epoxy monomer is known (for example, Patent Documents 1 and 2), epoxy monomers have a problem that they are difficult to handle, since, in general, they are liquid or gaseous at normal temperature and pressure, and are harmful to human health and environment, and also have a problem that it is necessary to allow a polyamide resin to react with an epoxy monomer in advance prior to kneading an ethylene-vinyl alcohol copolymer and a modified elastomer. In case that a polyamide resin and an epoxy monomer are reacted with each other in advance prior to kneading them with an ethylene-vinyl alcohol copolymer and a modified elastomer, a process for producing a thermoplastic elastomer composition is carried out in two steps of reacting the polyamide resin with the epoxy monomer, and kneading the resulting reaction product with the ethylene-vinyl alcohol copolymer and the modified elastomer, and therefore a more simple process for producing a thermoplastic elastomer composition has been required. In addition, in case that the reaction of a polyamide resin with an epoxy monomer is carried out using a twin-screw kneading extruder, it is difficult to knead a polyamide resin which is, with an epoxy monomer, since epoxy monomers are liquid or gaseous at normal temperature and pressure, and therefore it is necessary to introduce an epoxy monomer in the twin-screw kneading extruder from the middle of the kneading section of the twin-screw kneading extruder and to knead a polyamide resin in a molten state with the epoxy monomer. Accordingly, it is necessary to use a twin-screw kneading extruder having a high L/D ratio, a ratio of the effective screw length L to the screw diameter D. Twin-screw kneading extruders having a high L/D ratio are not practical, since they require a large facility investment and have a long residence time for materials within the extruder.